use the unit circle to determine the exact value of each angle: sine 30 degrees, cosine pi, sine 11 pi divided by 6, and cosine 135 degrees.
don't know how detailed your "unit circle" is , but you should easily read off that
sin30° = 1/2
cosπ = -1
sin 11π/6 = sin 330° = - sin30° = -1/2
cos 135 = -cos 45 = -√2/2
To determine the exact values of the given angles, we can use the unit circle. The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane.
To find the values of sine and cosine for any given angle, we look at the coordinates of the points where the terminal side of the angle intersects the unit circle.
1. Sine of 30 degrees: Start by converting 30 degrees to radians. To convert from degrees to radians, we multiply the degree measure by π/180. So, 30 degrees in radians is (30 * π) / 180 = π/6 radians. On the unit circle, the point corresponding to π/6 radians has coordinates (√3/2, 1/2). Since sine is the y-coordinate, the sine of 30 degrees is 1/2.
2. Cosine of π: Here, we need to convert π to degrees. Multiplying π by 180/π, we find that π radians is equal to 180 degrees. On the unit circle, the point corresponding to 180 degrees has coordinates (-1, 0). As cosine is the x-coordinate, the cosine of π radians is -1.
3. Sine of 11π/6: We can rewrite 11π/6 as 6π + π/6 radians. Since each full rotation around the unit circle adds 2π radians, we can disregard the 6π and focus on the remaining π/6. On the unit circle, the point corresponding to π/6 radians has coordinates (√3/2, 1/2). Hence, the sine of 11π/6 is 1/2.
4. Cosine of 135 degrees: To convert 135 degrees to radians, we multiply by π/180. Thus, 135 degrees in radians is (135 * π) / 180 = 3π/4 radians. On the unit circle, the point corresponding to 3π/4 radians has coordinates (-√2/2, √2/2). Hence, the cosine of 135 degrees is -√2/2.
So, the exact values of each angle are:
- Sine 30 degrees = 1/2
- Cosine π = -1
- Sine 11π/6 = 1/2
- Cosine 135 degrees = -√2/2